Assignment 1
COMP9418 – Advanced Topics in Statistical Machine Learning
Recall the guidance regarding plagiarism in the course introduction: this applies to this assignment, and if evidence
of plagiarism is detected, it will result in penalties ranging from loss of marks to suspension.
The dataset and breast cancer domain description in the Background section are from the assignment developed by
Peter Lucas, Institute for Computing and Information Sciences, Radboud Universiteit.
Introduction
In this assignment, you will develop some sub-routines in Python to implement operations on Bayesian Networks.
You will code an efficient independence test, learn parameters from complete data, and classify examples.
We will use a Bayesian Network for diagnosis of breast cancer. We start with some background information about
the problem.
Background
Breast cancer is the most common form of cancer and the second leading cause of cancer death in women. Every 1
out of 9 women will develop breast cancer in her lifetime. Although it is not possible to say what exactly causes
breast cancer, some factors may increase or change the risk for the development of breast cancer. These include
age, genetic predisposition, history of breast cancer, breast density and lifestyle factors. Age, for example, is the
most significant risk factor for non-hereditary breast cancer: women with age of 50 or older have a higher chance of
developing breast cancer than younger women. Presence of BRCA1/2 genes leads to an increased risk of developing
breast cancer irrespective of other risk factors. Furthermore, breast characteristics, such as high breast density are
determining factors for breast cancer.
The primary technique used currently for detection of breast cancer is mammography, an X-ray image of the breast.
It is based on the differential absorption of X-rays between the various tissue components of the breast such as fat,
connective tissue, tumour tissue and calcifications. On a mammogram, radiologists can recognise breast cancer by
the presence of a focal mass, architectural distortion or microcalcifications. Masses are localised findings, generally
asymmetrical to the other breast, distinct from the surrounding tissues. Masses on a mammogram are
characterised by several features, which help distinguish between malignant and benign (non-cancerous) masses,
such as size, margin, shape. For example, a mass with irregular shape and ill-defined margin is highly suspicious for
cancer, whereas a mass with round shape and well-defined margin is likely to be benign. Architectural distortion is
focal disruption of the normal breast tissue pattern, which appears on a mammogram as a distortion in which
surrounding breast tissues appear to be “pulled inward” into a focal point, often leading to spiculation (star-like
structures). Microcalcifications are tiny bits of calcium, which may show up in clusters, or in patterns (like circles or
lines) and are associated with extra cell activity in breast tissue. They can also be benign or malignant. It is also
known that most of the cancers are located in the upper outer quadrant of the breast. Finally, breast cancer is
characterised by several physical symptoms: nipple discharge, skin retraction, palpable lump.
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Breast cancer develops in stages. The early stage is referred to as in situ (“in place”), meaning that cancer remains
confined to its original location. When it has invaded the surrounding fatty tissue and possibly has spread to other
organs or the lymph, so-called metastasis, it is referred to as invasive cancer. It is known that early detection of
breast cancer can help improve the survival rates.
[20 Marks] Task 1 – Efficient d-separation test
In this part of the assignment, you will implement an efficient version of the d-separation algorithm. Let us start with
a definition for d-separation:
Definition. Let X, Y and Z be disjoint sets of nodes in a DAG G. We will say that X and Y are d-separated by Z, written
dsep(X,Z,Y), iff every path between a node in X and a node in Y is blocked by Z where a path is blocked by Z iff there
is at least one inactive triple on the path.
This definition of d-separation considers all paths connecting a node in X with a node in Y. The number of such
paths can be exponential. The following algorithm provides a more efficient implementation of the test that does
not require enumerating all paths.
Algorithm. Testing whether X and Y are d-separated by Z in a DAG G is equivalent to testing whether X and Y are
disconnected in a new DAG G0
, which is obtained by pruning DAG G as follows:
1. We delete any leaf node W from DAG G as long as W does not belong to X∪Y ∪Z. This process is repeated until
no more nodes can be deleted.
2. We delete all edges outgoing from nodes in Z.
Implement the efficient version of the d-separation algorithm in a function d_separation(G,X,Z,Y) that return a
boolean: true if X is d-separated from Y given Z and false otherwise.
[10 Marks] Task 2 – Estimate Bayesian Network parameters from data
Estimating the parameters of a Bayesian Network is a relatively simple task if we have complete data. The file bc.csv
has 20,000 complete instances, i.e., without missing values. The task is to estimate and store the conditional
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probability tables for each node of the graph. As we will see in more details in the Naive Bayes and Bayesian
Network learning lectures, the Maximum Likelihood Estimate (MLE) for those probabilities are simply the empirical
probabilities (counts) obtained from data.
Implement a function learn_outcome_space(data) that learns the outcome space (the valid values for each variable)
from the pandas dataframe data and returns a dictionary outcomeSpace with these values.
Implement a function learn_bayes_net(G, data, outcomeSpace) that learns the parameters of the Bayesian Network
G. This function should return a dictionary prob_tables with the all conditional probability tables (one for each node).
[20 Marks] Task 3 – Bayesian Network Classification
This particular Bayesian Network has a variable that plays a central role in the analysis. The variable BC
(Breast Cancer) can assume the values No, Invasive and InSitu. Accurately identifying its correct value would lead to
an automatic system that could help in early breast cancer diagnosis.
First, remove the variables metastasis and lymphnodes since these two variables can be understood as pieces of
information derived from BC and they may not be available at the point when BC is classified.
Use the Bayesian Network to classify cases of the dataset. First, use 10-fold cross-validation to split the dataset into
training and test sets. Use the function learn_bayes_net(G, data, outcomeSpace) to learn the Bayesian network
parameters from the training set.
Design a new function assess_bayes_net(G, prob_tables, data, outcomeSpace, class_var) that uses the test cases in
data to assess the performance of the Bayesian network. Implement the efficient classification procedure discussed
in the lectures. Such a function should return the classifier accuracy. Compute and report the average accuracy over
the ten cross-validation runs as well as the standard deviation.
[10 Marks] Task 4 – Naïve Bayes Classification
Implement a Naïve Bayes classifier. Design a new function assess_naive_bayes(G, prob_tables, data, outcomeSpace,
class_var) to classify the cases in data using the log probability trick discussed in the lectures. Do 10-fold crossvalidation,
same as above, and return accuracy and standard deviation. Since the Naïve Bayes classifier is essentially
a Bayesian network, you can call the function learn_bayes_net(G, data, outcomeSpace) to learn the Naïve Bayes
parameters from a training set.
[20 Marks] Task 5 – Tree-augmented Naïve Bayes Classification
Similarly to the previous task, implement a Tree-augmented Naïve Bayes (TAN) classifier and evaluate your
implementation in the breast cancer dataset. Design a function learn_tan_structure(data, outcomeSpace, class_var)
to learn the TAN structure (graph) from data and returns such a structure.
Since the TAN classifier is also a Bayesian network, you can use the function learn_bayes_net(G, data, outcomeSpace)
to learn the TAN parameters from a training set.
You can also use the previous designed function assess_bayes_net(G, prob_tables, data, outcomeSpace, class_var)
to classify and assess the test cases in data and measure the classifier accuracy.
[20 Marks] Task 6 – Report
Write a report (with less than 500 words) summarizing your findings in this assignment. Your report should address
the following:
a. Make a summary and discussion of the experimental results (accuracy). Use plots to illustrate your results.
b. Discuss the complexity of the implemented algorithms.
Use Markdown and Latex to write your report in the Jupyter notebook. Develop some plots using Matplotlib to
illustrate your results. Be mindful of the maximum number of words. Please, be concise and objective.